Given a differentiable tangent vector field v on a closed surface X with finitely many singularities p₁,…,p_n, Poincaré’s theorem states

where index(v;p_i) denotes the index of the singularity at p_i and χ‎(X) denotes the Euler charactertistic of X. The theorem implies the hairy ball theorem. Heinz Hopf later generalized the theorem to higher-dimensional manifolds.

A flow on a sphere

In the figure is a sphere with sink (index 1) at north pole and source (index 1) at south pole giving. χ‎ = 2.